Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

In this paper, we consider the problem of the existence of non-negative weak solution u of { u = 0 on ∂Ω Δu + up = 0 in Ω having a given closed set S as its singular set. We prove that when n/n - 2 < p < n + 2√n - 1/n - 4 + 2√n - 1 and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution u ∈ Ln+2/n-2 (Sn) of L0u + Ln+2/n-2 = 0 such that Sn is the singular set of u where L0 is the conformal Laplacian with respect to the standard metric of Sn. When n = 4 or 6, this kind of solution has been constructed by Pacard. © 1999 The Journal of Geometric Analysis.